Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{x^3 + 12x^2 + 20x}{-8x^2 - 48x + 320}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {x(x^2 + 12x + 20)} {-8(x^2 + 6x - 40)} $ $ p = -\dfrac{x}{8} \cdot \dfrac{x^2 + 12x + 20}{x^2 + 6x - 40} $ Next factor the numerator and denominator. $ p = - \dfrac{x}{8} \cdot \dfrac{(x + 10)(x + 2)}{(x + 10)(x - 4)}$ Assuming $x \neq -10$ , we can cancel the $x + 10$ $ p = - \dfrac{x}{8} \cdot \dfrac{x + 2}{x - 4}$ Therefore: $ p = \dfrac{ -x(x + 2)}{ 8(x - 4)}$, $x \neq -10$